Determination of Gravitational Potential at Ground Using Optical-Atomic Clocks on Board Satellites and on Ground Stations and Relevant Simulation Experiments

The general relativity theory provides a potential way to directly determine the gravitational potential (GP) difference by comparing the running rate or vibration frequencies of two optical-atomic clocks located at two stations. Recently we proposed an approach referred to as satellite frequency signal transmission based on the Doppler canceling technique or tri-frequency combination technique to determine the GP difference between a satellite and a ground site via exchanging microwave signals. Here, as an extension of our previous study, we aim to formulate determination of GP at ground stations and establish simulation experiments in different cases, including determining the GP at a ground station via one or more satellites and determining the GP difference between two ground stations via one or more satellites. Concerning each case we made simulating experiments, and results show that the precision of the GP at a ground station and that of the GP difference between two stations, determined via one satellite, are, respectively, about 0.383 and 0.454 m\(^{2}\)/s\(^{2}\), assuming the clocks with inaccuracy of about \(1\times 10^{-18}\) (s/s) level are available. If more satellites equipped with ultra-high-precise clocks are available, the precision of the determined GP (difference) at ground stations can be further improved.

Determination of Gravitational Potential at Ground Using Optical-Atomic Clocks on Board Satellites and on Ground Stations and Relevant Simulation Experiments

Determination of Gravitational Potential at Ground Using Optical-Atomic Clocks on Board Satellites and on Ground Stations and Relevant Simulation Experiments
Ziyu Shen 0 1
Wen-Bin Shen 0 1
Shuangxi Zhang 0 1
0 Key Lab of Surveying Engineering and Remote Sensing, Wuhan University , Wuhan , China
1 Department of Geophysics, School of Geodesy and Geomatics, Wuhan University , Wuhan , China
The general relativity theory provides a potential way to directly determine the gravitational potential (GP) difference by comparing the running rate or vibration frequencies of two optical-atomic clocks located at two stations. Recently we proposed an approach referred to as satellite frequency signal transmission based on the Doppler canceling technique or tri-frequency combination technique to determine the GP difference between a satellite and a ground site via exchanging microwave signals. Here, as an extension of our previous study, we aim to formulate determination of GP at ground stations and establish simulation experiments in different cases, including determining the GP at a ground station via one or more satellites and determining the GP difference between two ground stations via one or more satellites. Concerning each case we made simulating experiments, and results show that the precision of the GP at a ground station and that of the GP difference between two stations, determined via one satellite, are, respectively, about 0.383 and 0.454 m2/s2, assuming the clocks with inaccuracy of about 1 10 18 (s/s) level are available. If more satellites equipped with ultra-high-precise clocks are available, the precision of the determined GP (difference) at ground stations can be further improved.
Optical-atomic clocks; Microwave links; Tri-frequency combination; Satellite; Gravitational frequency shift; Gravitational potential determination
1 Introduction
According to the theory of general relativity (GR), an atomic (or optical atomic) clock’s
running rate and its vibration frequency will change at different positions with different
gravitational potentials (Einstein 1915; Weinberg 1972). Conversely, one can determine
the gravitational potential (GP) at a space point or on ground by measuring the change of
clocks’ running rates (Bjerhammar 1985) or by measuring the change of electromagnetic
signals’ frequencies (Shen et al. 1993). These alternative approaches of determining GP
(difference), referred to as the clock transportation comparison (CTC) and frequency signal
transmission comparison (FSTC), respectively, require clocks or oscillators with ultra-high
precision, say 1 10 18, which is equivalent to 1 cm in height. The time-frequency related
confirmation of the GR by various studies (Pound and Rebka 1959; Pound and Snider
1965; Hafele and Keating 1972; Vessot and Levine 1979; Turneaure et al. 1983; Chou
et al. 2010) provides a potential and prospective way to directly determine the GP based on
the CTC and FSTC.
In recent years, with quick development of high-precision clock manufacturing
technology, the optical-atomic clocks (OACs) with relative instability around 10 18 in several
hours and inaccuracy of 10 18 level have been generated in the laboratory (Hinkley et al.
2013; Bloom et al. 2014; Ushijima et al. 2015), and OACs with such precision level are
promising to be installed on satellites in the near future (Schiller et al. 2007; Tino et al.
2007). Since the current precision level of OACs is sufficient for applying the CTC or
FSTC approach to determining GP, it attracts more and more attention from geodesy,
geoscience and academia (Brumberg and Groten 2001; Pavlis and Weiss 2003; Bondarescu
et al. 2015). By far there are generally three kinds of methods that apply the GR to GP
determination: (1) transport clocks between two stations on ground and determine the GP
difference between the two stations by measuring the accumulated difference of the clocks
ticks (Bjerhammar 1985), (2) connect two stations by optical fiber or coaxial cable and
transmit frequency signals or time signals between the two stations (Shen and Peng 2012;
Shen 2013a, b; Shen and Shen 2015), (3) transmit frequency signals among different
stations on ground via GNSS-type (or communication-type) satellites (Shen et al.
1993, 2011; Shen 1998; Shen and Ning 2005).
Although the three kinds of methods mentioned above are all showing potential of
determining GP, the first two kinds have obvious drawbacks. For example, the clock
transportation comparison approach (Bjerhammar 1985) is laborious and time-consuming,
and the errors induced by transportation are difficult to control. The cable time transfer
approach (Shen and Shen 2015) or the fiber frequency transfer method (Shen and Peng
2012; Shen 2013a, b) is constrained by the distance between the two stations and increases
the complexity especially in the cases that we need to connect stations separated by ocean
and mountainous areas. Although the fiber frequency transfer comparison has reached
fairly high precision, about 10 19 level in relative accuracy (Grosche et al. 2009; Calonico
et al. 2014), the requirement of fibers limit its application in geodesy because we cannot
conveniently determine the GP at an arbitrary position. As contrast, the third kind of
method is most flexible and promising, since we can bridge any two places with one or
several satellites. It is less laborious, fast and unlimited to geography and distance.
The third method of the GP determination requires FSTC between ground and a
satellite. Currently, most relative experiments and researches aim to validate the
gravitational redshift effect predicted by the GR. Conversely, if the GR is proved to be reliable or
accurate enough, we can determine the GP based on the gravitational redshift effect. The
first ground-satellite frequency transfer experiment is the Gravity Probe A (GP-A)
experiment in 1976 (Vessot and Levine 1979), which aims to test the GR at 10 15 level in
frequency accuracy. And this experiment is the most precise direct test of the gravitational
redshift to date. Delva et al. (2015) proposed to test the gravitational redshift using Galileo
satellites with the frequency precision of 10 16, but when the experiments will be put into
practice remains uncertain. The next ground-satellite experiment similar to GP-A is the
future Atomic Clock Ensemble in Space (ACES) experiment planned to fly on the
International Space Station in 2017 (Cacciapuoti and Salomon 2011). The ACES will carry out
both the time and frequency transfer experiment to test the gravitational redshift, and the
precision of frequency transfer is supposed to be at 10 17 level. Furthermore, the
STEQUEST (Altschul et al. 2014) project, planned to launch in 2024, is proposed to equip an
optic-atomic clock with the stability of 1 10 18 in few hours to test the gravitational
redshift. Hence, the third method is very prospective in the near future.
Compared to time and frequency transfer on ground, the transfer between ground and a
satellite confronted much more problems and challenges. For example, the atmosphere and
ionosphere will cause signal delay and frequency shift, and the Earth rotation and tidal
effect also impose influence to the time and frequency transfer. When the accuracy
requirement of frequency transfer reaches 10 18 level or even higher, instead of the clock
stability, the systematic errors might be the dominant error sources. Wolf and Petit (1995)
detailedly studies in detail the clock synchronization in the vicinity of the Earth at the
accuracy level of 10 18. They analyzed the influence of various error sources, including
tidal effect, Doppler effect, external masses (Sun, Moon and other planets), atmosphere
pressure, polar motion and so on, and the error introduced by each of these factors is below
1 10 18 level after correction. But they did not consider the frequency shifts caused by
ionosphere and troposphere, which also need to be corrected. Ashby (1998) and Blanchet
et al. (2001) reexamined the GP-A test and improved the frequency transfer equation by
introducing the c 3 terms to the accuracy level of 5 10 17. They also analyzed the
influence of Shapiro time delay (Shapiro 1964) in frequency transfer. Considering the
GPA test, Linet and Teyssandier (2002) formulated a frequency shift in a gravitational field
generated by an axisymmetric rotating body and provided a one-way frequency transfer
equation accurate to c 4 terms, equivalent to the relative accuracy level higher than 1
10 18 in frequency. Shen et al. (2016) analyzed the ionosphere and troposphere influences
to the frequency links between a ground station and a GNSS-type satellite, and the
introduced errors can be reduced to 10 19 level after proper correction.
Thus, following our previous idea (Shen et al. 1993, 2011; Shen 1998; Shen and Ning
2005), we formulated an approach using frequency signals links based on the Doppler
canceling technique (DCT, see Kleppner et al. 1970; Vessot and Levine 1979) or
trifrequency combination (TrFC) technique to practically realize the determination of the GP
difference between a satellite and a ground station (Shen et al. 2016), which is referred to
as satellite frequency signal transmission (SFST). Based on our theoretical formulation, the
SFST can reach 1 m2/s2 if the clocks’ inaccuracy can achieve 1 10 17 level (Shen et al.
2016). Here we will extend the study of Shen et al. (2016) to the accuracy level of 10 18,
focusing on determining the GP at ground at centimeter level and conduct relevant
simulation experiments to show how to realize the GP determination based on the SFST.
2 Gravitational Potential Difference Determination Between a Satellite
and a Ground Site
Referring to Fig. 1, the SFST contains three microwave links (Shen et al. 2016). An emitter
at a ground station P emits a frequency signal fe at time t1. When the signal is received by a
satellite S at time t2, it immediately transmits the received signal fe0 and emits a frequency
signal fs at the same time. These two signals transmitted and emitted from the satellite are
received by a receiver at the ground station P at time t3. During the period of the emitting
and receiving, the position of the ground station in space has been changed from P to P0
(see Fig. 1).
Based on the procedures as described above (also see Fig. 1), we can extract the gravity
frequency shift signals (or equivalently gravitational frequency shift signals). Suppose we
set a basic frequency f0 and fe ¼ fs ¼ f0, then the frequency shift signals can be determined
as depicted in Fig. 2. The frequencies of the signals emitted from ground oscillator and
satellite oscillator are f0. The microwave link 1 and link 2 consist a go-return link by a
phase-coherent microwave transponder equipped at the satellite and provide two-way
frequency shift data as a beat frequency f000 f0 (Shen et al. 2016). Similarly, the
microwave link 3 provides one-way frequency shift data as a beat frequency f00 f0 (Shen et al.
2016). The output frequency Df is expressed as (Kleppner et al. 1970; Vessot and Levine
1979; Vessot et al. 1980; Shen et al. 2016):
In free space, the output frequency expressed by Eq. (1) implies that it can completely
cancels the first-order Doppler effect. This is the reason that this procedure is referred to as
Doppler canceling technique (DCT, see, e.g., Vessot and Levine 1979). Hence, the GP
difference between the satellite and the ground site can be obtained from the following
equation (Vessot and Levine 1979):
Here an Earth-centered inertial coordinate frame has been applied in Eq. (3), where /s and
/e are Newtonian GPs at spacecraft (or satellite) and ground station, respectively, ve and vs
are velocities of ground station and spacecraft, respectively, rse is vector from spacecraft to
Fig. 1 Ground station P emits a
frequency signal fe at time t1.
Satellite S transmits the received
signal fe0 and emits a new
frequency signal fs at time t2. The
ground station receives signals fe00
and fs0 at time t3 at position P0. /
is GP, r is position vector, v is
velocity vector, a is centrifugal
acceleration vector (modified
after Shen et al. 2016)
Spacecraft S
Fig. 2 Ground oscillator emits a frequency signal f0 to the spacecraft (or satellite), then the spacecraft
transmits the received signal to ground and emits a frequency signal f0 from spacecraft oscillator to the
ground at the same time (modified after Vessot and Levine 1979; Shen et al. 2016)
ground station, ae is centrifugal acceleration vector of ground station, c refers to the speed
of light in vacuum.
Equation (2) has omitted the terms higher than c 2, and it holds only at the accuracy
level or a little better than 10 15 (Cacciapuoti and Salomon 2011). For a higher precision
requirement, Ashby (1998) and Blanchet et al. (2001) appended the c 3 terms to Eq. (2)
and established an equation suitable for an accuracy level of 5 1017. However, to achieve
an accuracy level of 1 10 18, terms up to c 4 should be considered. A theoretical
formula of one-way frequency transfer in free space accurate to 1 10 18 was given by
Linet and Teyssandier (2002). Based on the study of Linet and Teyssandier (2002), with
three-link frequency transmission as described in Fig. 1, Eq. (1) can be expressed as
(derived in the Appendix in detail)
where the frequency shift ‘‘output’’ Df is given by expression (1), / is the Earth’s
Newtonian GP, D/es ¼ /s /e is the GP difference between satellite and ground station,
qðiÞði ¼ 1; 2; 3; 4Þ are referred to Eqs. (50) and (29)–(32), and their explanations are
provided thereafter. Equation (3) takes a different form from Eq. (2) because the latter aims
only to the accuracy level of c 2 (see Vessot and Levine 1979), while the former keeps all
terms accurate to c 4.
We note that, based on the Doppler canceling technique (see Fig. 2), an oscillator
(clock) with stability of 10 18 is necessary to control its emitting frequency. Then, by
trifrequency combination we may cancel out the Doppler effect and precisely draw out the
GP difference between a ground station and the spacecraft. This is the reason why a precise
clock on board a spacecraft is needed.
Equation (3) has included the effect of Shapiro delay and the effect of the axisymmetric
rotating body of the Earth. It can reach the accuracy of 10 19 level, but holds only in free
space. In real space outside the Earth, a signal’s frequency will be contaminated by
ionospheric and tropospheric effects and other influences (Shen et al. 2016), which means
that certain corrections should be further appended to the equation. In addition, the
potential difference /se contains the influences of other factors (such as tidal effect,
gravitational potential fields of celestial bodies). If we aim to obtain the GP difference
caused by the Earth, such influences should also be removed. The residual errors after all of
the corrections, together with the systematic errors (such as equipment errors, orbit
uncertainty), are the main factors that determine the final accuracy of the frequency
transfer based on the tri-frequency combination (TrFC) technique.
When various corrections and influences are taken into consideration, Eq. (3) is
modified as the following equation
where Kf is the sum of all correction terms, df is the sum of all error terms.
The correction term Kf in Eq. (4) is expressed as
Kf ¼ Kfion þ Kftro þ Kftide þ Kfceles
where Kfion and Kftro are, respectively, the corrections of ionospheric and tropospheric
effects (after tri-frequency combination), Kftide is the contribution of the additional
potential associated with the Earth’s deformation caused by tidal effects, Kfceles is caused
by the GP generated by the main celestial members in our solar system, including the Sun,
the Moon and other planets. Accordingly, after the corrections as expressed as (5), the total
residual errors are expressed as
Thus the error terms df in Eq. (4) can be expressed as
where dfsys is the sum of all relevant systematic errors, which will be discussed later.
The correction terms Kfion and Kftro have been studied in detail in Shen et al. (2016),
expressed as
dfcor ¼ dfion þ dftro þ dftide þ dfceles
where H is the height (in km) of the spacecraft from the ground, q is the average electron
density (in m-3), M1 and M2 are substitutions for simplification, defined as
M1 ¼ 77:6 10 6p=T ; M2 ¼ 0:373e=T 2, and M1 and M2 are the average value of M1 and
M2 along the signals’ propagation paths (see Shen et al. 2016), where p; T ; e are,
respectively, total pressure (in mbar), temperature (in degrees K), and partial pressure of water
vapor (in mbar); ae is the acceleration vector of the ground station. The magnitude of
correction terms Kfion and Kftro and their residual errors after corrections are listed in
Table 2 (which is explained later).
The deformation of Earth will cause the potential changes outside the Earth and the
position changes on the surface of the Earth. These two effects consist of the correction
term Kftide, which are expressed in spherical harmonics expansion series (Farrell 1972).
The tide-induced potential changes in the free space are most conveniently modeled as
variations in the standard geopotential coefficients Cnm and Snm (Eanes et al. 1983), and
their contributions can be estimated from some global tide models (e.g., Parke 1982). They
can also be calculated by some mature softwares (Tsoft for example, see Camp and
Vauterin 2005), and the residual error is at the millimeter level.
Concerning the last correction term Kfceles, the GP influence of each planet [except for
the Sun, which is expressed as Eq. (11)] can be expressed as:
þ Oi; ði ¼ Moon; Mercury; Venus; . . .Þ
where G is gravitational constant, Mi is the mass of the celestial body, ri is the vector from
a celestial body to the ground station, rse is the vector from the ground station to the
satellite. Oi is the higher-order potentials caused by non-spherical distribution. However,
Eq. (10) is not suitable for the Sun because of the equivalence principle (Kleppner et al.
1970; Hoffmann 1961). The GP influence of the Sun should be expressed as (Hoffmann
1961):
rsat ic
r2
c
rgrd ic
r2
c
where rc is the vector from Sun to the Earth’s mass center, ic is the unit vector of rc, rsat
and rgrd are, respectively, the vectors of the satellite and ground station with respect to the
Earth’s mass center. Because of the long distances and relatively small gravitational
influences, we can omit the Oi terms in Eqs. (10) and (11). Estimations show that Oi does
not exceed 10 3 m2/s2, equivalent to 10 20 in frequency influence (see Table 1). The
ephemeris of solar system planets can be obtained from Ephemerides of Planets and the
Moon (Pitjeva 2013), and the errors of planets’ orbit determination are negligible in our
estimation (even an orbit offset of 1 km for the Moon causes a frequency error at the level
of 10 20, and for the other planets are even smaller). For the potential difference
measurement between a GNSS-type satellite and a ground station, the largest correction
magnitude of each celestial body (when the body, the satellite and the ground station are
located in one straight line) is listed in Table 1. We can see that to achieve the accuracy
level of 1 10 18 for measuring the Earth’s GP difference, all of other celestial bodies
except for Neptune need to be considered. Then, after the celestial bodies’ corrections, the
residual errors dfceles are below 1 10 20 (Table 2).
The error term dfsys in Eq. (7) is caused by all the effects that cannot be properly or
effectively modeled and corrected, such as the equipment delays, clock errors, satellite’s
orbit errors. Here we denote dfsys as
dfsys ¼ dfvepo þ dfdelay þ dfosc þ dfo
where the relevant terms are explained in what follows.
dfvepo is the position and velocity errors of ground station and satellite. The position
error in the precise ephemeris of a GPS satellite is about 10 2 m (Kang et al. 2006; Guo
et al. 2015), and the velocity error can be reduced to below 10 5 m/s (Sharifi et al. 2013).
Table 1 Largest correction magnitude of each celestial body for the GP measurement between a GNSS
satellite and a ground station
Correction magnitude (relative frequency shift)
The position error of a ground station is negligible because it is relatively small compared
to a satellite. Then, the errors introduced from position vector can be estimated by applying
error propagation to Eq. (3), and the amount of dfvepo is below 3:4 10 19 (see Table 2). In
addition, Duchayne et al. (2009) have studied the influence of position difference between
reference point and mass center of a satellite and concluded that the introduced error is
totally negligible.
dfdelay is introduced by all the equipment delays. Notice that the DCT method performs
frequency transfer, without involving time transfer, thus the hardware time delays in both
ground station and satellite can be neglected. But satellite’s transponder delay must be
taken into consideration, because the satellite is in motion, its position when receiving
signals is different from that when emitting signals. For a transponder’s delay at about
800 ns (Pierno and Varasi 2013), its introduced error dfdelay is at the level of 10 19 (Shen
et al. 2016).
dfosc is oscillator (clock) error. Since our aim is to determine the GP difference to
accuracy of centimeter level, which means that clocks at least with stability and accuracy
of 10 18 level are required. In this paper we assume that the clocks meet our requirement
(some day in the near future), and their instability can achieve 1 10 18 in an hour
(dfosc\10 18). It should be noted that such high-precise optical-atomic clock has been
realized in the laboratory (Bloom et al. 2014). Although currently the stablest clocks
onboard satellites are at 10 17 stability level (Cacciapuoti and Salomon 2011), the stability
of 10 18 level (onboard satellites) will be achieved in the near future. And since an atomic
clock is sensitive to temperature and magnetic field (see, e.g., Rochat et al. 2012), it is also
important to stabilize the inner environment of a satellite to minimize the introduced errors.
In this paper, we do not discuss these influences because it is a topic of clock
manufacturing, and different clocks might vary in sensitivity to temperature and magnetic field.
Finally, the term dfo denotes all of the higher-order contributors (multi-path effects,
polar motion, etc.) that can be safely neglected. The magnitudes of all correction terms and
error terms are listed in Table 2. We can see that the magnitudes of some of the error
sources are different from those given by Wolf and Petit (1995), due to the fact that we
focus on frequency comparison between satellite and ground links, while they studied the
Table 2 Error magnitudes of different error sources in determining GP difference between a satellite and a
ground station
(Residual) Error magnitudes
a This error is estimated based on Eq. (1). The error sources come from three measurement processes,
including that the ground station P emits a frequency signal at time t1, the spacecraft (satellite) S receives
and transmits signals at time t2, and the ground station receives the signals at time t3 at position P0 (see
Fig. 1). According to Eq. (1), f ¼ f00 f0 f0002 f0 ¼ f00 þ f0 2f000, we have r2 ¼ r002 þ ðr02 þ r0002Þ=4 ¼ r0þ
2r20=4 ¼ 3r20=2. Then, we have the error magnitude r ¼ ð1:73=1:41Þr0 1:23 10 18, where we take
r0 ¼ 1:0 10 18
clock synchronization at ground or satellite. And some of error estimates in Wolf and Petit
(1995) are undetailed (for example, the influence of celestial bodies).
It should be noted that the DCT method is designed for microwave links, because the
measurements concern frequency comparison. To our knowledge, in free or medium space
it is not suitable for optical links, which are mainly used for time transfer (e.g., laser time
transfer). In addition, there are some constraints for the frequency of f0. A higher value of
f0 helps to reduce the influence of the ionosphere [see Eq. (8)] and reduce the refraction of
propagation path. However, if the f0 value is too high ([30 GHz for example), the energy
required for sustaining the system greatly increases, and the signal will strongly attenuated
by the Earth’s atmosphere and particles contained in it, especially during wet weather. In
practice, a frequency band range from 2 GHz (adopted in the GP-A experiment) to 15 GHz
(adopted in the ACES mission) is suitable.
3 Determination of GP Difference Between Two Ground Sites
The SFST approach (Shen et al. 2016) was designed for determining the GP difference
between a satellite and a ground site, as described in Sect. 2. However, if two ground sites
are connected to the same satellite via satellite links simultaneously, the satellite can serve
as a ‘‘bridge’’ to connect the two ground sites (Shen et al. 1993, 2016). Thus the GP
difference between the two ground sites can be determined. Figure 3 depicts the concept.
The ground stations P1 and P2 simultaneously observe a satellite S, which is at the same
time visible by these two ground stations.
According to Eq. (3), for each of the ground stations P1 and P2, we have the following
equations:
Fig. 3 Links of frequency
signals among a satellite and two
ground stations. The satellite
S receives frequency signals from
two ground stations P1 and P2
simultaneously and then
transmits the signals back to
ground stations. The ground
stations receives the transmitted
frequency signals at P01 and P02
because of Earth rotation
where the subscripts 1 and 2 denote, respectively, the values related to stations P1 and P2.
Combining Eqs. (13) and (14), we obtain the equation which contains the GP difference
between the two ground stations:
where /e21 ¼ /e1 /e2 is the Newtonian GP difference between the two ground stations
P1 and P2. The error term df12 is the sum of the errors df1 and df2. From Eq. (15) we can
see that since there is a pair of SFST links in determining the GP difference between two
ground sites, the error magnitude from most error sources would be larger compared to one
SFST link. However, the error sources from satellite, such as the error caused by velocity
and position, can be significantly reduced because of their partial cancelations as shown by
Eq. (15). They cannot be completely canceled out because although we intend to establish
a pair of SFST links to one satellite simultaneously, in reality it is not quite possible to link
two different ground stations at exactly the same time, because (1) even if the clocks
located at two stations have been a prior synchronized, there may still exist time difference
and (2) the signals propagation paths between the satellite and the stations are different (see
Fig. 3). If the satellite receives the two signals from ground stations P1 and P2 at slightly
different instants t1 and t2, the velocities of the satellite (vs and v0s at times t1 and t2) as
shown in Eqs. (13) and (14) will be different. Thus the error term df12 in Eq. (15) contains a
new error source which comes from asynchronism:
where dfasy is the asynchronism error, and the meanings of the other terms are the same as
described in Eq. (7). In order to estimate the magnitude of dfasy, suppose the time interval
between the received two signals from two ground stations is Dt, and in the time duration
Dt the satellite’s velocity changed from vs to v0s. Then, we have:
where as is centrifugal acceleration vector of the satellite. Substituting the vs to v0s in
Eq. (14), and then combining it to Eqs. (13) and (15), we can obtain the expression of dfasy:
where Oðc 3Þ are small amounts of (and higher than) c 3 terms. With Eq. (18) we can
estimate the influence of dfasy. For example, suppose the satellite-receiving time difference
Dt ¼ 1 ms, and the satellite is a GPS satellite whose centrifugal acceleration jasj is about
0.558 m/s2, velocity jvsj is about 3000 m/s (Zhang et al. 2006). Notice that vs and as are
almost orthogonal, then the calculated value of dfasy is below 10 19, which is negligible in
our case.
In order to guarantee that the satellite-receiving time difference Dt \ 1 ms or even
smaller, we can employ the following two techniques. (1) According to the orbit of
satellite, we can preestimate the distances between the satellite and the two ground stations
(e.g., the distances of P1S and P2S in Fig. 3). Then, we can determine the suitable time for
emitting frequency signals from the two ground stations. (2) When the satellite receives the
frequency signals from the two ground stations, it can send a feedback signal that contains
the information of the current satellite-receiving time difference. Once the two ground
stations receive the feedback signals, they can adjust the signals’ emitting times
correspondingly. We note that, as mentioned in Sect. 2, exact time synchronization is not
necessary, because we make frequency transfer.
4 Simulation Experiments
Sections 2 and 3 provide the approaches of determining the GP difference in two cases: GP
difference determination between one satellite and one ground station (Shen et al. 2016)
and GP difference determination between two ground stations. In practical applications,
these approaches can be used flexibly. For example, to improve the accuracy of the results,
we can determine the GP at a certain ground station or the GP difference between two
ground stations via several satellites (spacecrafts) which are equipped with high-precision
clock systems. For the purpose of potential applications of the SFST approach in GP
measurements in the future, in this section we conducted several simulation experiments as
examples.
4.1 The Error Models of Various Error Sources
The reliability of a simulation experiment depends on whether the simulating case is close
to the real case. In our experiments, we use GPS satellites whose orbit data are obtained
from IGS product Web site (www.igs.org/products), and two ground stations located in
China whose coordinates are also given. The key problem is the simulation of various error
effects as described in Sect. 2 and 3. Because although the magnitude of each error source
has been estimated, it is difficult to predict the value of each error in continuous
experiments. In order to solve the problem, we adopt three kinds of error models in accordance
with the different natures of the error sources.
First, the state of high-precision atomic clocks should be properly simulated. Galleani
et al. (2003) have developed a mathematical model for clock error which can be expressed
as:
X2ðtÞ ¼ c2 þ at þ r2/2ðtÞ
where t 0 represents time, X1 represents the phase deviation, X2 represents the frequency
deviation, x(0) and y(0) are initial conditions of X1 and X_ 1, respectively, /1ðtÞ and /2ðtÞ
are Wiener processes (Brownian motion) defined by dW ðtÞ ¼ nðtÞdt, where nðtÞ is a white
Gaussian noise with zero mean. r1 and r2 are constants that represent the diffusion
coefficients of the two noises, a is a drift term, c2 is the initial condition of X2ðtÞ.
According to Eq. (19), a series of simulated clock data with errors embedded can be
generated.
Second, we consider the error model of the satellite orbit. In the local satellite frame, the
position errors of a satellite have three scalar components Dx, Dy and Dy. In this local
coordinate system (x, y, z), x points to the normal axis of the orbit plane, y points to the
tangential axis, z points to the radial axis. According to Hill model, the velocity errors of a
satellite satisfy the following equations (Colombo 1986):
XDx0 sin Xt þ Dx_0 cos Xt
2Dz_0 sin Xt þ ð4Dy_0 þ 6XDz0Þ cos Xt
ð3Dy_0 þ 6XDz0Þ
where t 0 represents time, the subscript ‘‘0’’ denote the initial condition, X is the orbital
angular frequency. It should be noted that Eq. (20) holds in a rotating coordinate system
(rotates about the x axis). For a non-rotating frame whose axes coincide with the moving
ones at time t, there holds the following transformation:
Dx_ðtÞ ¼ Dx_ðtÞNR
Dy_ðtÞ ¼ Dy_ðtÞNR
Dz_ðtÞ ¼ Dz_ðtÞNR þ XDyðtÞ
where the subscript ‘‘NR’’ means ‘‘non-rotating’’.
Finally, for other error sources (see Table 2), currently there are no mature
mathematical models to simulate. Thus we adopted a general function of Wiener process to
represent each of the other error sources:
where X(t) is the error value at time t, /ðtÞ and nðtÞ are both standard white Gaussian
noises, a, b and c are constant coefficients. Clearly Eq. (22) is a simplified model and
cannot perfectly simulate the values of various error sources. However, taking into
consideration the large number of error sources and the relatively small amount of their
magnitudes, this simplification is acceptable in our simulation experiments that aims at
testing the precision of the SFST approach.
In summary, for each error source we have assigned an error model: Eq. (19) for the
clock errors, Eq. (20) for satellite position and velocity errors and Eq. (22) for each of other
error sources. The error models are independent of each other, and the coefficient values in
a certain equation are determined in accordance with the error magnitude of the relevant
error source. Then, the final error model of our simulation experiment is a combination of
all these error models. In the following subsections, we conduct four types of simulation
experiments and provide the corresponding results. The parameters and relevant error
magnitudes used in our simulation experiments are listed in Tables 2 and 3.
4.2 Determination of the GP at a Ground Station Via a Satellite
One of the presently most accurate Earth gravitational model [e.g., EGM2008 (Pavlis et al.
2012a)] provides an accuracy about 10–20 cm (equivalent to 2 m2/s2 in potential) at
ground and may achieve at least 0.1 m2/s2 level at the (GNSS-type) target satellite altitude,
which is around 20,000 km above the geoid. Hence, here in this study we just assume that
the gravitational potential at the orbit of a GNSS-type or communication-type satellite is
given at the accuracy level of 0.1 m2/s2, which is equivalent to 1 cm in height in the
domain near the Earth’s surface.
We choose a ground station in Wuhan, China, whose geodetic coordinate is 114:32 E,
30:52 W, 50 m. The observation time period is 2.5 h, from 10:30 a.m. to 1:00 p.m.,
January 31, 2016. In this relatively short time duration, we choose a nearest GPS satellite
(PG27) for our simulation experiments. At the ground station, the angle between the
observation sight and zenith is 50:16 at start point. It first decreases then increases to
30:77 at end point during our experiments, as schematically shown by Fig. 4. Here we use
simulation experiments via one satellite links to determine the GP at the Wuhan ground
station based on the SFST and analyze the accuracy of the results. The simulation
experiment method and relevant results are described as follows.
First, the orbit information of the GPS satellite PG27 is obtained from IGS product Web
site (www.igs.org/products). The precise ephemeris contains position and clock
information, the time interval between two data set being 15 min. However, our simulation
experiments are conducted every 10 s, hence the required data set was obtained by
interpolation. The orbit data and ground position are regarded as true value, and we use
EGM2008 model (Pavlis et al. 2012a) to calculate the GP values at ground station and
Table 3 Relevant parameters used in simulation experiments
GP error of satellite
Additional parameters can be referred to Table 2
satellite orbit at different times. These GP values are also regarded as true values. Other
parameters such as the velocities of ground station and satellite can be calculated. The
electron density (ionosphere influence) can also be obtained from IGS, and the atmosphere
condition (troposphere influence) can be obtained from Earth Global Reference
Atmospheric Model (Leslie and Justus 2011). Then, the ionosphere and troposphere residual
corrections Kfion and Kftro can be calculated from Eqs. (8) and (9). The obtaining of other
correction terms (Kftide, Kfposition and Kfceles) has been illustrated in Sect. 2. These
correction terms are all regarded as true values. Thus according to Eq. (4), we can calculate
the true value of the output frequency Df =f0.
The next step is adding noises. In Sect. 4.1, we have discussed the three kinds of error
models, and the noises are generated according to these models and then added to the
relevant true values. Consequently, we get a new set of ‘‘observations’’ which are used to
estimate the value of interest. Then, we use Eq. (4) to calculate the GP difference /Si /Ei
at time ti and denote it as Di. Taking equal weight of each observation (at time ti), we have
Pn
i¼1 ð/Si
n
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pin¼1 /Si Di /E 2
where /E and r/E are the mean value of the estimated GPs at the ground station and the
corresponding standard deviation (SD), respectively, /Si is the GP (with noises added) at
the satellite orbit at time ti, n is the total number of the ‘‘observations’’.
By comparing the estimated value and the true value at time ti (i ¼ 1; 2; . . .; n), we may
verify the reliability of our proposed SFST approach (Shen et al. 2016). The results are
shown in Fig. 5a. We can see that most of the absolute offset values are below the order of
1 m2/s2. There are 900 observations in total, and the mean value of the differences between
Fig. 5 Gravitational potential (GP) of the ground station in Wuhan determined by: a the GPS satellite
PG27, b 5 GPS satellites in combination (PG27, PG26, PG23, PG16 and PG08). And the gravitational
potential (GP) differences between the two ground stations in Wuhan and Nanjing, determined by: c the
GPS satellite PG27, d 5 GPS satellites in combination (PG27, PG26, PG23, PG16 and PG08). Experiment
time period is from 10:30 to 13:00, January 31, 2016. We have an ‘‘observation’’ every 10 s and compare the
true value with the estimated value. There are 900 comparisons for each satellite and the offsets between
true values and estimated values are drawn as time series
the estimated GP and the true value at ground station at time ti ði ¼ 1; 2; . . .; nÞ is
-0.383 m2/s2, and the corresponding standard deviation (SD) is 0.385 m2/s2. The results
in details are listed in Table 4 as Case 1.
4.3 Determination of GP at a Ground Station Via Observing Several
Satellites
If more GNSS-type satellites equipped with high-precise clocks are available, the results of
determining the GP at the ground station could be improved.
The setup of our second simulation experiment is similar to the first one as described in
Sect. 4.2. The experiment date, time duration and location of ground station remain
unchanged. The difference is that we use 5 GPS satellites (PG08, PG16, PG23, PG26 and
PG27) to establish SFST links to the ground station at Wuhan. They are the most nearest
GPS satellites in the experimental time period, and all of the angles among the observation
sights and zeniths do not exceed 65 . For each set of links between one satellite and the
ground station, the procedures are as same as described in Sect. 4.2, and we obtain one
estimate of the GP at the ground station via every satellite. Taking different weights based
on the separated accuracies, we obtain the weighted results, expressed as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi
/E ¼ Pj5¼P1 5/j /Ej ; r/E ¼ uuutPj5¼1P5/j r2/Ej ð24Þ
j¼1 /j j¼1 /j
where j denotes the jth satellite, /j denotes the weight of the results based on the satellite j,
/Ej and r/Ej denote the estimated GP at ground station and the corresponding accuracy
Table 4 Setup and results of simulation experiments of four cases
62,555,817.884 62,555,817.501
62,555,817.884 62,555,817.612
2911.615 2912.069
2911.615 2911.905
In case 1 we calculate the GP at a ground station using one satellite. In case 2 we calculate the GP at a
ground station via 5 different satellites. In case 3 we calculate the GP difference of two ground stations via
one satellite. In case 4 we calculate the GP difference of two ground stations via 5 different satellites. The
GP at satellite position is calculated based on the Earth gravitational model EGM2008
based on the jth satellite. Then /E and r/E are the final results by combining the
measurements of 5 satellites. Without obvious difference, here we just take equal weight, then
the results are shown in Table 4 as case 2, and Fig. 5b shows the offset between the
estimated values and the true values at time ti. We can see that the result is better than (a),
because some of the error sources can be reduced by multiple measurements. Here there
are 900 observations for each satellite, and 4500 observations in total. The mean value of
the differences is -0.272 m2/s2, and the standard deviation (SD) is 0.216 m2/s2.
If a satellite is connected with two ground stations via the SFST links simultaneously, as
shown in Fig. 6, the GP difference between these two ground stations can be measured
according to the results of the two groups of the SFST links. In this case, although the error
sources from the satellite, such as the error of velocity and position can be significantly
reduced, there exist new error sources stemming from the satellite-receiving simultaneity
problem (see Sect. 3). Suppose two ground stations A and B are linked to a same satellite as
link LA and LB. The measurement times of LA are tAi (i ¼ 1; 2; . . .; N), and the measurement
times of LB are tBi (i ¼ 1; 2; . . .; N). These measurement times are recorded by the clock
(time-keeping system) equipped on the satellite; thus, they share the same time standard.
We use the error models of Eq. (22) to simulate the asynchronism error, and the magnitude
of dfasy is below 10 19 as shown in Eq. (18).
The setup of our experiments here is very similar to the first experiment as described in
Sect. 4.2. The difference is that we added another ground station which is located in
Nanjing (about 500 km from Wuhan station), with its geodetic coordinate being 118:78 E,
32:05 W, 30 m. The theoretical formulation is referred to Sect. 3. Since the two ground
sties (Wuhan station A and Nanjing station B) can be bridged by one satellite or multiple
satellites simultaneously, we made experiments corresponding to different cases.
For the mentioned two ground stations connected by one satellite, we have:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uutPin¼1 D/^ASi D/^BSi D/AB 2
Fig. 6 Experiments are conducted at the time duration when satellite moves from position S to position S0
(from 10:30 a.m. to 1:00 p.m., January 31, 2016). Two ground stations move from P1 and P2 to P01 and P02
where D/AB and rD/AB are the mean value and standard deviation (SD) of the estimated GP
difference between stations A and B, respectively, D/^ASi and D/^BSi are the estimated
(calculated after adding noises) values of the GP differences between satellite and ground
stations A and B at time ti, respectively. n denotes the total number of the ‘‘observation
pairs’’, and there are 900 pairs of measurements in this case.
Similarly, for two ground stations connected by multiple satellites (here as an example
we use 5 satellites), we have:
where j denotes the jth satellite, /j denotes the weight of the jth satellite. Since there are 5
satellites in total, D/ðjÞAB and rD/ðjÞAB denote the determined results based on the jth
satellite. Then D/AB and rD/AB are the final results by combining the results based on 5
satellites. Taking equal weight, the results are shown in Table 4 (see cases 3 and 4), and
Fig. 5c, d. In case 3 (see Fig. 5c) we use one satellite (PG27) to connect two ground
stations; the mean value of the differences is 0.454 m2/s2, and standard deviation (SD) is
0.567 m2/s2. In case 4 (see Fig. 5d) we use 5 different satellites (PG08, PG16, PG23,
PG26, PG27) to connect the two ground stations simultaneously (these 5 satellites are
visible at the same time for the two ground stations in the experiment time duration), the
mean value of the differences is 0.290 m2/s2, and standard deviation (SD) is 0.504 m2/s2.
We can see that the results as shown by Fig. 5d are a little better than those as shown by
Fig. 5c, but not very obviously. This is because some kinds of errors (such as the clock
errors of ground stations, the tidal correction residual errors) cannot be obviously reduced
by simply combining multiple satellites.
5 Conclusions
As a further improvement of the study of Shen et al. (2016), in this paper we formulated an approach
for determining the GP of a ground station and the GP difference between two ground stations via
one or more satellites and provided various simulating experiments addressing four different cases
based on the tri-frequency combination (TrFC) technique. The precisions of determining the
absolute GP of a ground station and the GP difference between two ground stations are estimated,
reaching the level of 0.1 m2/s2, as long as the clocks’ stability and inaccuracy achieve the level
of 1 10 18. Various influence factors (such as the tidal effects, the potentials of other
celestial bodies, the frequency influences along the propagation path) have been considered
and estimated. Their introduced errors do not exceed the error magnitude of clocks.
In Sect. 4 we have discussed the SFST approach of determining the GP of a ground
station given one or several satellites’ GPs. Inversely, given GPs at ground stations with
proper distribution, we can determine the GP at the orbit of a flying satellite. One potential
application of the SFST approach is to determine the GP distribution along one or several
GOCE-type satellites orbits and provide a potential distribution over a quasi-spherical
surface constructed by the GOCE-type satellites. To complete this potential and
prospective task, we need to first establish a ground datum station network to cover the
whole orbits of the GOCE-type satellites via the SFST approach. The core idea is similar to
determining the absolute GP at ground stations as described in this paper. Details are
discussed in a separate study. Currently, due to the fact that optical clocks with stability of
10 18 level have been successfully realized (Poli et al. 2014; Bongs et al. 2015), with very
quick development of time and frequency science, in the near-future portable optical
clocks with stability of 10 18 level could be also realized. Consequently, the SFST
approach will be prospective and potential for determining GP at any space position, not
only providing an alternative approach for directly determining the GP, but also providing
a way to realize the unification of the world height datum system.
Acknowledgements We sincerely thank three anonymous Reviewers and Prof. Michael Rycroft for their
valuable comments and suggestions, which greatly improved the manuscript. This study is supported by National
973 Project China (Grant Nos. 2013CB733301, 2013CB733305), NSFC (Grant Nos. 41210006, 41374022,
41429401), DAAD (Grant No. 57173947) and NASG Special Project Public Interest (Grant No. 201512001).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
Appendix: Formulation of Tri-Frequency Combination for Frequency
Transfer at Accuracy Level of 10218 in Free Space
For the purpose of an accuracy level of 10 18 in frequency transfer, the terms of c 4 should
be taken into consideration. In this Appendix we will derive a formula for the tri-link
frequency transmission measurement [see Fig. 1; Eq. (3) in Sect. 2] in free space, achieving
the accuracy requirement of 10 18 level.
where qðiÞ is in the order of 1=ci (i ¼ 1; 2; 3; 4), expressed as
/ and W are, respectively, the first and second Newtonian potentials, defined as
G Z
W ¼ c2
jqx ðxx0Þ0j c þ 12 v2 þ ð1 2bÞ/ þ P þ 3c qp d3x0 ð35Þ
where q is the rest mass density, P is the specific energy density (ratio of internal energy
density to rest mass density), p is the pressure, q is the conserved mass density, given by
and f is vector potential, defined as
where x is the Earth’s temporal rotation angular velocity. Here in this study, for our
purpose we just take x as a constant vector in what follows.
Accurate to our requirement, lAðxA; xBÞ and lBðxA; xBÞ are expressed as (Linet and
Teyssandier 2002)
NAB þ ½lMðxA; xBÞ þ lJ2 ðxA; xBÞ þ flSðxA; xBÞ þ lvr ðxA; xBÞg
½lMðxB; xAÞ þ lJ2 ðxB; xAÞ þ flSðxB; xAÞ þ lvr ðxB; xAÞg
GMJ2
lJ2 ðxA; xBÞ ¼ ðc þ 1Þ c2
where C is the Earth’s principal moment of inertia around z axis, r denotes the Euclidean
norm of the vector x, RAB ¼ xB xA, RAB ¼ jRABj, k is the unit vector along positive z
axis, n ¼ x=r is the unit vector along x direction (example, nA ¼ xA=rA). In Eqs. (31) and
(32)
lðB2Þ=c2 ¼
lJ2 ðxB; xAÞ; lðB3Þ=c3 ¼ lSðxB; xAÞ þ lvr ðxB; xAÞ;
where the relevant vectors are given by expressions (40)–(43).
Suppose the emitting frequency at ground station A at time t1 is f0, the receiving
frequency at satellite at time t2 is f00, similar to Eq. (28), we have
fact qðAiðÞ1ÞBð2Þ). This received signal at satellite (at time t2) is immediately transponded at
time t2 to the ground station, and the receiving frequency f 00 at ground station at time t3 is,
based on Eq. (28), expressed as
Simultaneously at time t2 the satellite emits a new signal with frequency fs, and the
receiving frequency at ground is, based on equation (28), expressed as
Concerning Eq. (47), it is worthy to notice that we omitted the difference between
transponding the received signal at time t2 and emitting a new signal at time t20 at satellite,
and that of the receiving signals at time t3 and time t30 at ground, namely we assume that
t20 ¼ t2, t30 ¼ t3. This is related to time synchronization, which is not so serious in our
frequency transfer scheme, because we compare the frequency, not the time elapsed.
Now we formulate a tri-frequency combination based on the emitting frequencies f0 and
fs at ground and satellite, respectively, and the observed receiving frequencies f000 and fs0.
We examine the following frequency shift ‘‘output’’
Substituting Eqs. (45)–(47) into Eq. (48), we obtain the following expression
where qðiÞ is defined as
the fact that in a very short period (\0.5 s in general cases for our present purpose), we
may consider the GP and speed of the ground station hold invariant. However, if necessary
at any time, we may just apply the expression following the first equal sign ‘‘¼’’ of
Eq. (49).
We note that all relevant quantities are related to t1; t2, or t3. For instance, lðAiÞ and lðBiÞ are
related to t1; t2, or t3. Based on Eq. (49) one can determine the GP difference between
A and B.
The value of qðiÞ can be calculated by the given velocities of A and B and proper model
values of the Newtonian potentials /, W and the vector potential f. We use the GP model
EGM2008 (Pavlis et al. 2012b) to calculate the model value of /, denoted as /EGM08,
which has at least the accuracy levels of tens of centimeters (one meter height is equivalent
to 10 m2/s2 potential) at ground station and several centimeters at the satellite altitude.
/EGM08 is a harmonic expansion expression of the Earth’s external GP complete to degree/
order 2159. Hence, in Eqs. (31) and (32) we use the following model value
In practice, for the purpose of model value used here, it is accurate enough to use the terms
up to degree/order 20 of the EGM2008 model.
level of c 4, in Eq. (32) we may take the following model value
Concerning the vector potential f, from expressions (37) and (32) we see that it also
plays only a role of the order 1=c4. Hence, to achieve the accuracy level of 1
Eq. (32) we may take the following model value
In our simulation experiment, we just set b ¼ 1=2; c ¼ 1; a1 ¼ 0.
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